Tycho Brahe and Johannes Kepler · Johannes Kepler The Music of the Spheres SC/STS 3760, VIIISC/STS...

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TychoTycho Brahe and Brahe and Johannes KeplerJohannes Kepler

The Music of the SpheresThe Music of the Spheres


TychoTycho BraheBrahe15461546--16011601

Motivated by Motivated by astronomy's predictive astronomy's predictive powers.powers.Saw and reported the Saw and reported the NovaNova of 1572.of 1572.Considered poor Considered poor observational data to be observational data to be the chief problem with the chief problem with astronomy.astronomy.


TychoTycho Brahe at Brahe at UraniborgUraniborg

Established an Established an observatoryobservatory----UraniborgUraniborgon on HvenHven, an island off , an island off Denmark.Denmark.

Worked there 20 years.Worked there 20 years.Became very unpopular Became very unpopular with the local residents.with the local residents.

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TychoTycho’’ss ObservationsObservations

Made amazingly precise Made amazingly precise observations of the observations of the heavens with nakedheavens with naked--eye eye instruments.instruments.Produced a huge globe Produced a huge globe of the celestial sphere of the celestial sphere with the stars he had with the stars he had identified marked on it.identified marked on it.


TychoTycho, the Imperial Mathematician, the Imperial MathematicianLeft Left UraniborgUraniborg to become the Imperial to become the Imperial Mathematician to the Holy Roman Mathematician to the Holy Roman Emperor at the Court in Prague.Emperor at the Court in Prague.TychoTycho believed that Copernicus was correct believed that Copernicus was correct that the planets circled the Sun, but could that the planets circled the Sun, but could not accept that the Earth was a planet, nor not accept that the Earth was a planet, nor that it moved from the centre of the that it moved from the centre of the universe.universe.He developed his own compromise system.He developed his own compromise system.


TychoTycho’’ss SystemSystem

Earth stationary.Earth stationary.Planets circle Sun.Planets circle Sun.Sun circles Earth.Sun circles Earth.Problem: Problem:

Could not get Could not get MarsMarsto fit the system.to fit the system.Note the Note the intersecting paths intersecting paths of the Sun and of the Sun and Mars that bothered Mars that bothered Copernicus.Copernicus.

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Johannes KeplerJohannes Kepler

15711571--16301630

LutheranLutheranMathematics professor in Mathematics professor in Austria (Graz)Austria (Graz)Sometime astrologerSometime astrologerPythagorean/NeoPythagorean/Neo--PlatonistPlatonistOne of the few One of the few Copernican convertsCopernican converts


Pythagorean/Platonic regularities in Pythagorean/Platonic regularities in the Heavensthe Heavens

Why are there precisely 6 planets in the heavens Why are there precisely 6 planets in the heavens (in the Copernican system)?(in the Copernican system)?

Mercury, Venus, Earth, Mars, Jupiter, SaturnMercury, Venus, Earth, Mars, Jupiter, SaturnWith a Pythagorean mindset, Kepler was sure there With a Pythagorean mindset, Kepler was sure there was some mathematically necessary reason.was some mathematically necessary reason.He found a compelling reason in Euclid.He found a compelling reason in Euclid.

A curious result in solid geometry that was attributed to A curious result in solid geometry that was attributed to Plato.Plato.


Euclidean Regular FiguresEuclidean Regular Figures

A regular figure is a closed linear figure with every side and every angle equal to each other.

•For example, an equilateral triangle, a square, an equilateral pentagon, hexagon, and so forth.

There is no limit to the number of regular figures with different numbers of sides.

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Inscribing and CircumscribingInscribing and Circumscribing

All regular figures can be inscribed within a circle and also circumscribed around a circle.The size of the figure precisely determines the size of the circle that circumscribes it and the circle that is inscribed within it.


Regular SolidsRegular Solids

In three dimensions, the comparable constructions are In three dimensions, the comparable constructions are called regular solids.called regular solids.They can inscribe and be circumscribed by spheres.They can inscribe and be circumscribed by spheres.


The Platonic SolidsThe Platonic Solids

Unlike regular figures, their Unlike regular figures, their number is not unlimited. number is not unlimited. There are actually only There are actually only five possibilities:five possibilities:

Tetrahedron, Cube, Tetrahedron, Cube, Octahedron, Dodecahedron, Octahedron, Dodecahedron, IcosahedronIcosahedron

This was discussed by Plato. They are traditionally called the This was discussed by Plato. They are traditionally called the ““Platonic Solids.Platonic Solids.””That there could only be five of them was proved by Euclid in That there could only be five of them was proved by Euclid in the last proposition of the last book of the last proposition of the last book of The Elements.The Elements.

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KeplerKepler’’s brainstorms brainstorm

Kepler imagined that (like Kepler imagined that (like EudoxeanEudoxean spheres), spheres), the planets were visible dots located on the the planets were visible dots located on the surface of nested spherical shells all centered on surface of nested spherical shells all centered on the Earth.the Earth.There were six planets, requiring six spherical There were six planets, requiring six spherical shells. Just the number to be inscribed in and shells. Just the number to be inscribed in and circumscribe the five regular solids. circumscribe the five regular solids.


Six Planets, Five SolidsSix Planets, Five Solids

Like Pythagoras, Kepler believed that neat Like Pythagoras, Kepler believed that neat mathematical relationships such as this could mathematical relationships such as this could not be a coincidence. It must be the key to not be a coincidence. It must be the key to understanding the mystery of the planets.understanding the mystery of the planets.There were six planets because there were five There were six planets because there were five Platonic solids. The Platonic solids. The ““spheresspheres”” of the planets of the planets were separated by the inscribed solids. Thus were separated by the inscribed solids. Thus their placement in the heavens is also their placement in the heavens is also determined.determined.


The Cosmographical MysteryThe Cosmographical Mystery

In 1596, Kepler published a In 1596, Kepler published a short book, short book, MysteriumMysteriumCosmographicumCosmographicum, in which he , in which he expounded his theory.expounded his theory.The 6 planets were separated The 6 planets were separated by the 5 regular solids, as by the 5 regular solids, as follows:follows:

Saturn / Saturn / cube cube / Jupiter / / Jupiter / tetrahedron tetrahedron / Mars / / Mars / dodecahedron dodecahedron / Earth / / Earth / icosahedron icosahedron / Venus / / Venus / octahedronoctahedron / Mercury/ Mercury

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What was the mystery?What was the mystery?

The cosmographical mystery that Kepler The cosmographical mystery that Kepler ““solvedsolved”” with with the Platonic solids was the provision of reasons why the Platonic solids was the provision of reasons why something that appeared arbitrary in the heavens something that appeared arbitrary in the heavens followed some exact rule. This is classic followed some exact rule. This is classic ““saving the saving the appearancesappearances”” in Platoin Plato’’s sense.s sense.The arbitrary phenomena were:The arbitrary phenomena were:

The number of planets.The number of planets.Their spacing apart from each other.Their spacing apart from each other.

Both were determined by his arrangement of spheres Both were determined by his arrangement of spheres and solids.and solids.


Kepler and Kepler and TychoTycho BraheBrahe

Kepler's cosmic solution didn't exactly work, but Kepler's cosmic solution didn't exactly work, but he thought it would with better data.he thought it would with better data.TychoTycho had the data.had the data.Meanwhile Meanwhile TychoTycho needed someone to do needed someone to do calculations for him to prove calculations for him to prove hishis system.system.A meeting was arranged between the two of A meeting was arranged between the two of them.them.


Kepler, the Imperial MathematicianKepler, the Imperial Mathematician

Kepler became Kepler became Tycho'sTycho's assistant in 1600.assistant in 1600.TychoTycho died in 1601.died in 1601.Kepler succeeded Kepler succeeded TychoTycho as Imperial as Imperial Mathematician to the Holy Roman Emperor in Mathematician to the Holy Roman Emperor in Prague, getting all of Prague, getting all of Tycho'sTycho's data.data.

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Kepler's Kepler's ““DiscoveriesDiscoveries””

Kepler found many magical and mysterious Kepler found many magical and mysterious mathematical relations in the stars and planets.mathematical relations in the stars and planets.He published his findings in two more books:He published his findings in two more books:

The New AstronomyThe New Astronomy, 1609, 1609The Harmony of the WorldThe Harmony of the World, 1619, 1619

Out of all of this, three laws survive.Out of all of this, three laws survive.The first two involve a new shape for astronomy, the The first two involve a new shape for astronomy, the ellipse.ellipse.


Conic SectionsConic Sections

In addition to Euclid, Kepler would have known of the work of the Hellenistic mathematician Apollonius of Perga, who wrote a definitive work on what are called conic sections: the intersection of a cone with a plane in different orientations. Above are the sections Parabola, Ellipse, and Hyperbola.


The EllipseThe Ellipse

The Ellipse is formed by a plane cutting completely through the The Ellipse is formed by a plane cutting completely through the cone.cone.Another way to make an ellipse is with two focal points (A and BAnother way to make an ellipse is with two focal points (A and B above), and above), and a length of, say, string, longer than the distance AB. If the sta length of, say, string, longer than the distance AB. If the string is stretched ring is stretched taut with a pencil and pulled around the points, the path of thetaut with a pencil and pulled around the points, the path of the pencil point is pencil point is an ellipse. In the diagram above, that means that if C is any poan ellipse. In the diagram above, that means that if C is any point on the int on the ellipse, AC+BC is always the same.ellipse, AC+BC is always the same.

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Kepler's first lawKepler's first law

1. The planets travel in 1. The planets travel in elliptical orbits with the elliptical orbits with the sun at one focus.sun at one focus.

All previous astronomical All previous astronomical theories had the planets theories had the planets travelling in circles, or travelling in circles, or combinations of circles.combinations of circles.Kepler has chosen a Kepler has chosen a different geometric figure.different geometric figure.


A radical idea, to depart from circlesA radical idea, to depart from circles

Kepler’s ideas were very different and unfamiliar to astronomers of his day.


What was the mystery?What was the mystery?

KeplerKepler’’s s ““first lawfirst law”” gives some account of the actual gives some account of the actual paths of the planets (i.e., paths of the planets (i.e., ““savessaves”” them).them).All of the serious astronomers before him had found All of the serious astronomers before him had found that simple circular paths didnthat simple circular paths didn’’t quite work. Ptolemyt quite work. Ptolemy’’s s EarthEarth--centered system had resorted to arbitrary centered system had resorted to arbitrary epicycles and epicycles and deferentsdeferents, often off, often off--centre. Copernicus centre. Copernicus also could not get circles to work around the sun.also could not get circles to work around the sun.Kepler found a simple geometric figure that described Kepler found a simple geometric figure that described the path of the planets around the sun. the path of the planets around the sun.

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Kepler's second lawKepler's second law

2. A radius vector from the sun to a planet sweeps out 2. A radius vector from the sun to a planet sweeps out equal areas in equal times. equal areas in equal times.


What is the mystery here?What is the mystery here?

The second law provides a mathematical relationship The second law provides a mathematical relationship that accounts for the apparent speeding up of the that accounts for the apparent speeding up of the planets as they get nearer the sun and slowing down as planets as they get nearer the sun and slowing down as they get farther away.they get farther away.Kepler had no explanation Kepler had no explanation whywhy a planet should speed up a planet should speed up near the sun. (He speculated that the sun gave it some near the sun. (He speculated that the sun gave it some encouragement, but didnencouragement, but didn’’t know why.) t know why.) But in Platonic fashion he provided a formula that But in Platonic fashion he provided a formula that specifies the relative speeds.specifies the relative speeds.


Kepler's third lawKepler's third law

3. The Harmonic Law: 3. The Harmonic Law: dd 33//tt 22 = k= kThe cube of a planetThe cube of a planet’’s mean distance s mean distance dd from the sun from the sun divided by the square of its time divided by the square of its time tt of revolution is the of revolution is the same for all planets.same for all planets.That is, the above ratio is equal to the same constant, That is, the above ratio is equal to the same constant, k, for all planets.k, for all planets.

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The mystery cleared up by the third The mystery cleared up by the third lawlaw

Kepler noted that the planets all take different times to Kepler noted that the planets all take different times to complete a full orbit of the Sun. complete a full orbit of the Sun. The farther out a planet was from the Sun, the longer The farther out a planet was from the Sun, the longer was its period of revolution.was its period of revolution.He wanted to find a single unifying law that would He wanted to find a single unifying law that would account for these differing times.account for these differing times.The 3The 3rdrd law gives a single formula that relates the law gives a single formula that relates the periods and distances of all the planets.periods and distances of all the planets.

As usual, Kepler did not provide a cause for this relationship.As usual, Kepler did not provide a cause for this relationship.


Kepler's three laws at a glanceKepler's three laws at a glance

1. The planets travel in elliptical orbits with the sun at 1. The planets travel in elliptical orbits with the sun at one focus.one focus.

Accounts for the orbital paths of the planets.Accounts for the orbital paths of the planets.

2. A radius vector from the sun to a planet sweeps out 2. A radius vector from the sun to a planet sweeps out equal areas in equal times. equal areas in equal times.

Accounts for the speeding up and slowing down of the Accounts for the speeding up and slowing down of the planets in their orbits.planets in their orbits.

3. The Harmonic Law: 3. The Harmonic Law: dd 33//tt 22 = k= kAccounts for the relative periods of revolution of the planets, Accounts for the relative periods of revolution of the planets, related to their distances from the sun.related to their distances from the sun.


Why these are called KeplerWhy these are called Kepler’’s lawss laws

Kepler did not identify these three statements about the Kepler did not identify these three statements about the behaviour of the planets as his behaviour of the planets as his ““laws.laws.””

We call these KeplerWe call these Kepler’’s laws because Isaac Newton pulled s laws because Isaac Newton pulled them out of Keplerthem out of Kepler’’s works and gave Kepler credit for them.s works and gave Kepler credit for them.

Kepler found many Kepler found many ““lawslaws””——meaning regularities about meaning regularities about the heavensthe heavens——beginning with the cosmographical beginning with the cosmographical mystery and the 5 Platonic solids. mystery and the 5 Platonic solids.

Most of these we ignore as either coincidences or error on his Most of these we ignore as either coincidences or error on his part.part.

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What did Kepler think he was doing?What did Kepler think he was doing?

Kepler has all the earmarks of a Pythagorean. Kepler has all the earmarks of a Pythagorean. A full and complete explanation is nothing more nor A full and complete explanation is nothing more nor less than a mathematical relationship describing the less than a mathematical relationship describing the phenomena. phenomena. In AristotleIn Aristotle’’s sense it is a formal cause, but not an s sense it is a formal cause, but not an efficient, nor a final cause.efficient, nor a final cause.


The Music of the SpheresThe Music of the Spheres

As a final example of KeplerAs a final example of Kepler’’s frame of mind, s frame of mind, consider the main issue of his last book, consider the main issue of his last book, The The Harmony of the World.Harmony of the World.KeplerKepler’’s goal was to explain the harmonious s goal was to explain the harmonious structure of the universe. structure of the universe. By harmony he meant the same as is meant in By harmony he meant the same as is meant in music.music.


Music of the Spheres, 2Music of the Spheres, 2

Since Pythagoras it has been known that a Since Pythagoras it has been known that a musical interval has a precise mathematical musical interval has a precise mathematical relationship. Hence all mathematical relations, relationship. Hence all mathematical relations, conversely, are musical intervals.conversely, are musical intervals.If the planetsIf the planets’’ motions can be described by motions can be described by mathematical formula, the planets are then mathematical formula, the planets are then performing music.performing music.

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In particular, the orbits of the planets, as they In particular, the orbits of the planets, as they move through their elliptical paths, create move through their elliptical paths, create different ratios, which can be expressed as different ratios, which can be expressed as musical intervals.musical intervals.The angular speeds at which the planets move The angular speeds at which the planets move determine a pitch, which rises and falls through determine a pitch, which rises and falls through the orbit.the orbit.



As follows:As follows:Mercury, a scale running Mercury, a scale running a tenth from C to Ea tenth from C to EVenusVenus——almost a perfect almost a perfect circular orbitcircular orbit——sounds sounds the same note, E, the same note, E, through its orbit.through its orbit.Earth, also nearly Earth, also nearly circular, varies only from circular, varies only from G to AG to A--flatflat



Mars, which has a more Mars, which has a more irregular path than Venus or irregular path than Venus or the Earth, goes from F to C the Earth, goes from F to C and back.and back.Jupiter moves a mere minor Jupiter moves a mere minor third from G to Bthird from G to B--flat.flat.Saturn moves a major third Saturn moves a major third from G to B.from G to B.The Moon too plays a tune, The Moon too plays a tune, from G to C and back.from G to C and back.

Tycho Brahe and Johannes Kepler · Johannes Kepler The Music of the Spheres SC/STS 3760, VIIISC/STS...

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